Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 270467, 8 pageshttp://dx.doi.org/10.1155/2013/270467

Research ArticleBinary Tree Pricing to Convertible Bonds with Credit Riskunder Stochastic Interest Rates

Jianbo Huang,1 Jian Liu,2 and Yulei Rao1

1 School of Business, Central South University, Changsha, Hunan 410083, China2 School of Economics & Management, Changsha University of Science & Technology, Changsha 410004, China

Correspondence should be addressed to Yulei Rao; yuleirao@sina.com

Received 18 January 2013; Accepted 21 March 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Jianbo Huang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The convertible bonds usually have multiple additional provisions that make their pricing problem more difficult than straightbonds and options. This paper uses the binary tree method to model the finance market. As the underlying stock prices and theinterest rates are important to the convertible bonds, we describe their dynamic processes by different binary tree. Moreover, weconsider the influence of the credit risks on the convertible bonds that is described by the default rate and the recovery rate; thenthe two-factor binary tree model involving the credit risk is established. On the basis of the theoretical analysis, we make numericalsimulation and get the pricing results when the stock prices are CRR model and the interest rates follow the constant volatility andthe time-varying volatility, respectively. This model can be extended to other financial derivative instruments.

1. Introduction

Convertible bonds with the characteristic of bonds and stockare a complex financial derivative. They provide the rightto holders that they give up the future dividend to obtainsome stock with specified quantity.The pricing of convertiblebonds is more difficult than straight bonds and options;the main reason is that it not only has value of bonds, butalso involves all kinds of embedded option value broughtby provisions of conversion, call, put, and so on. What ismore, the embedded options in most times are Americanoptions. So, generally speaking, the pricing of convertiblebonds cannot get closed-form solution; in most conditions,numerical method was adopted, for example, binary treemethod, Monte Carlo method, finite difference method, andso on. As for Monte Carlo method, firstly it uses differentstochastic differential equations to describe the pricing factormodels in the market for simulation, then it makes pricingbased on the characteristic of convertible bonds, for example,the boundary conditions acquired by all kinds of provisions,to make pricing for conversion bonds (Ammann et al. [1],Guzhva et al. [2], Kimura and Shinohara [3], Yang et al. [4],and Siddiqi [5]). But because the supposed parameters of

stochastic differential equations are exogenous, this methodnot necessarily makes a better fitting for the existing marketconditions. The binary tree method can solve the aboveproblems.

The binary tree method is firstly put forward by Cox etal. [6], Cox-Ross-Rubinstein (CRR) binomial option pricingmodel. After that, many researchers revised and popularizedit. Cheung and Nelken [7] firstly apply binary tree to convert-ible bonds pricing and obtain the pricing solution of a two-factormodel which is based on stock prices and interest rates.Carayannopoulos and Kalimipalli [8] apply the trigeminaltree pricingmodel to convertible bonds pricing research withsingle factor. Hung andWang [9] also apply binary treemodelto convertible bonds pricing which embodies default risk andconsiders the influences of stock prices and interest rates.Chambers and Lu [10] further considered the correlation ofstock prices and interest rates and expanded the model ofHung and Wang. Binary tree model has been widely usedin the pricing of contingent claims such as stock options,currency options, stock index options, and future options. Xu[11] proposes a trinomial lattice model to price convertiblebonds and asset swaps with market risk and counterpartyrisk.

2 Abstract and Applied Analysis

Interest rate is a very important factor in the financialmarket; all security prices and yields are related to it. Theinterest rates model has equilibrium model, no-arbitragemodel, and so on. In the equilibrium model, interest ratesare generally described by some stochastic process, whichis mean-reversion, to make interest rates show the trend ofconvergence to a long-term average with the passing of time,including Vasicek model, Rendleman-Bartter model, andCIR model. The parameters of these models should be esti-mated with history data, by selecting parameters purposely,but generally this fitting is not accurate and even reasonablefitting formula cannot be found. No-arbitrage models makeinitial term structure as model input and construct a binarytree such as CRR model for interest rate process, so that theterm structure can fit the reality better andmore concise.Therelatively wide application no-arbitrage models include Ho-leemodel, Hull-Whitemodel, Black-Derman-Toymodel, andHeath-Jarrow-Mortonmodel. Different from the interest ratemodel of Hung and Wang and Chambers and Lu, this paperuses constant volatility and time-varying volatility binary treemodel to describe short interest rateswhich aremore intuitiveand convenient.

This paper makes pricing research of convertible bondswith the call and put provisions and uses binary tree methodfor the modeling of state variables in the financial market.As the duration of convertible bonds is relatively longerthan straight bonds, their prices are subject to the impactof interest rates. Moreover, as one of the corporate bonds,convertible bonds may have credit risk. So this paper usesdifferent binary trees to model the process of stock prices andinterest rates, and considering the impact of stock dividendsand credit risk to convertible bonds, it adopts default rateand recovery rate to describe credit risk and get two-factorbinary tree model involving credit risk; on this basis, withexample simulation to get the convertible bonds, pricingresults under the condition of stock price obey CRR model,constant volatility interest rate binary tree model, and time-varying volatility interest rate binary tree model.

2. Market Model

2.1. Interest Rates Binary Tree

2.1.1. Constant Volatility Binary Tree Model. Ritchken [12]deduces the continuous form of short-term 𝑟(𝑡) in Ho-Lee model [13] meeting the following stochastic differentialequations:

𝑑𝑟 (𝑡) = 𝜇 (𝑡) 𝑑𝑡 + 𝜎 (𝑡) 𝑑𝑧 (𝑡) , 𝑡 > 0, (1)

where𝜇(𝑡) is the drift,𝜎(𝑡) is the instantaneous volatility, bothcan be the function of time 𝑡, and 𝑧(𝑡) is Brownian motion.

Grant and Vora [14] get the discrete form of (1) as follows:

Δ𝑟 (𝑡) = 𝜇 (𝑡) Δ𝑡 + 𝜎 (𝑡) Δ𝑧 (𝑡) , 𝑡 ≥ 0. (2)

Make𝑓(𝑗) to be the forward interest rate in the interval [𝑗, 𝑗+1]. Then get

𝜇 (0) Δ𝑡 = 𝑓 (1) − 𝑟 (0) +

Δ𝑡

2

𝜎2(𝑟 (1)) ,

𝜇 (1) Δ𝑡 = 𝑓 (2) − 𝑓 (1) +

Δ𝑡

2

𝜎2(

2

∑

𝑗=1

𝑟 (𝑗)) − Δ𝑡 ⋅ 𝜎2(𝑟 (1)) ,

𝜇 (𝑡 − 1) Δ𝑡 = 𝑓 (𝑡) − 𝑓 (𝑡 − 1) +

Δ𝑡

2

𝜎2(

𝑡

∑

𝑗=1

𝑟 (𝑗))

− Δ𝑡 ⋅ 𝜎2(

𝑡−1

∑

𝑗=1

𝑟 (𝑗)) +

Δ𝑡

2

𝑡−2

∑

𝑛=1

𝜎2(

𝑛

∑

𝑗=1

𝑟 (𝑗)) ,

∀𝑡 ≥ 3,

𝑡

∑

𝑛=0

𝜇 (𝑛) Δ𝑡 = 𝑓 (𝑡 + 1) − 𝑟 (0) +

𝑡

∑

𝑛=1

𝛿 (𝑛) , ∀𝑡 ≥ 1,

(3)

where

𝑡

∑

𝑛=0

𝛿 (𝑛) =

Δ𝑡

2

𝜎2(

𝑡+1

∑

𝑗=1

𝑟 (𝑗)) −

Δ𝑡

2

𝜎2(

𝑡

∑

𝑗=1

𝑟 (𝑗)) ,

∀𝑡 ≥ 1,

𝜎2(

𝑡

∑

𝑗=1

𝑟 (𝑗)) = 𝜎2(

𝑡

∑

𝑗=1

(𝑡 − 𝑗 + 1) 𝜎𝑗−1

Δ𝑧𝑗−1

)

=

𝑡

∑

𝑗=1

(𝑡 − 𝑗 + 1)2

𝜎2

𝑗−1Δ𝑡.

(4)

Suppose that volatility is constant; that is, 𝜎(𝑡) = 𝜎𝑐, ∀𝑡 >

0, and then

𝜎2(

𝑡

∑

𝑗=1

𝑟 (𝑗)) = 𝜎2(

𝑡

∑

𝑗=1

(𝑡 − 𝑗 + 1) 𝜎𝑐Δ𝑧𝑗−1

)

= 𝜎2

𝑐

𝑡

∑

𝑗=1

(𝑡 − 𝑗 + 1)2

Δ𝑡.

(5)

And get the constant volatility interest rates binary tree asshown in Figure 1.

2.1.2. Time Varying Volatility Binary Tree Model. Jarrow andTurnbull [15] supposed that the volatility of short-term ischangeable in different intervals, but is constant in the sametime interval. Let Δ𝑡 = 1, and then the discrete form ofinterest rates can meet

𝑟 (𝑡) = 𝑟 (0) +

𝑡−1

∑

𝑗=0

𝜇 (𝑗) + 𝜎 (𝑡 − 1)

𝑡−1

∑

𝑗=0

Δ𝑧 (𝑗) . (6)

Abstract and Applied Analysis 3

𝑟0

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝑟0 + 𝜇0Δ𝑡 + 𝜎𝑐√Δ𝑡

𝑟0 + 𝜇0Δ𝑡 − 𝜎𝑐√Δ𝑡

𝑟0 +1∑𝑗=0𝜇𝑗Δ𝑡 + 2𝜎𝑐√Δ𝑡

𝑟0 +1∑𝑗=0𝜇𝑗Δ𝑡

𝑟0 +1∑𝑗=0𝜇𝑗Δ𝑡 − 2𝜎𝑐√Δ𝑡

𝑟0 +2∑𝑗=0𝜇𝑗Δ𝑡 + 3𝜎𝑐√Δ𝑡

𝑟0 +2∑𝑗=0𝜇𝑗Δ𝑡 + 𝜎𝑐√Δ𝑡

𝑟0 +2∑𝑗=0𝜇𝑗Δ𝑡 − 𝜎𝑐√Δ𝑡

𝑟0 +2∑𝑗=0𝜇𝑗Δ𝑡 − 3𝜎𝑐√Δ𝑡

Figure 1: 4-period constant volatility interest rates binary tree.

The variance formula of the sum of short-term interest rate is

𝜎2(

𝑡

∑

𝑗=1

𝑟𝑗) =

𝑡−1

∑

𝑗=0

(𝑗 + 1) 𝜎2

𝑗+

𝑡−2

∑

𝑗=0

𝑡−1

∑

𝑘=𝑗+1

2 (𝑗 + 1) 𝜎𝑗𝜎𝑘. (7)

And then get time-varying interest rates tree as shown inFigure 2.

2.2. Stock Price Binary Tree. Suppose that the currentmoment is 0 and the expiration date of convertible bonds isT. According to interval Δ𝑡, we divide the period [0, 𝑇] to 𝐿subintervals: [𝑡

𝑖, 𝑡𝑖+1], 0 ≤ 𝑖 ≤ 𝐿, 𝑡

0= 0, 𝑡

𝐿= 𝑇, 𝑇 = 𝐿Δ𝑡.

In each interval [𝑡𝑖, 𝑡𝑖+1], there are two possible states in the

market, up or down. The change of every market state isindependent. U means the up state and D means the downstate.

The stock prices will have two states; 𝑝 means theprobability of market up, and then the probability of marketdown is 1 − 𝑝. If the current stock price is 𝑆, then the stockprice of later period may have two possibilities: 𝑆

𝑢, 𝑆𝑑, and

𝑆𝑢= 𝑆 × 𝑢, 𝑆

𝑑= 𝑆 × 𝑑; 𝑢, 𝑑 separately mean the magnitude

of up and down. If the initial price of stock is known, thenthe stock price tree can be determined by the given modelparameters 𝑝, 𝑢, and 𝑑.

Model parameters 𝑝, 𝑢, and 𝑑 will directly impact theresults of binary tree; the selection of them should followno-arbitrage principle. Generally speaking, there are twoselections: CRRmodel [4], equal-probability binomial model(Roman [16], Hull [17]). This paper adopts CRR model todescribe pricing process of stock.

CRR model selects parameters as follows:

𝑢 = 𝑒𝜎𝑠√Δ𝑡

, 𝑑 = 𝑢−1= 𝑒−𝜎𝑠√Δ𝑡

,

𝑝 =

1

2

[1 +

𝜇𝑠

𝜎𝑠

√Δ𝑡] .

(8)

Especially, if the actual financial market is changed to riskneutral market. Then the expected profit 𝜇

𝑠of stock will

change to risk-free interest rate 𝑟, but the volatility 𝜎𝑆is the

same. The probability of price up in this model is 𝑝 = (𝑒𝑟Δ𝑡

−

𝑑)/(𝑢−𝑑); among them, 𝑟 is risk-free interest rate. Under thiscondition, the pricing result is no arbitrage.

2.3. Credit Risk. Consider convertible bonds with creditrisk. We adopt the method of Jarrow and Turnbull [18] tomodel the credit risk of convertible bonds. Suppose that theprobability of default risk in time interval [𝑡

𝑖−1, 𝑡𝑖] is𝜆𝑖and the

rate of recovery is 𝜉𝑖when default. If there are serials different

deadline risk-free zero-coupon bonds in the financial marketand the prices are {𝑃(1), 𝑃(2), 𝑃(3), . . . , 𝑃(𝑛)}, the serials ofdifferent deadline risk company zero-coupon bonds and theprices are {𝐷(1), 𝐷(2), 𝐷(3), . . . , 𝐷(𝑛)}. We can get the risk-free interest term structure and risk interest term structurefrom them. If the recovery rate 𝜉

𝑖is already known, then the

rate of risk 𝜆𝑖, in number 𝑖 period of bonds, can be acquired.

The detail analysis process is as follows.If the risk-free interest rate of one-year period is 𝑟

0and

risk interest rate is 𝑟∗1, then

𝑒−𝑟∗

1= [1 ⋅ (1 − 𝜆

1) + 𝜉1⋅ 𝜆1] 𝑒−𝑟0

, and get 𝜆1=

1 − 𝑒𝑟0−𝑟∗

1

1 − 𝜉1

.

(9)

If the risk interest rate of two-year period is 𝑟∗2, then

𝑒−2𝑟∗

2= {[1 (1 − 𝜆

2) + 𝜉2𝜆2] 𝑒−𝑟𝑢

𝜋 (1 − 𝜆1)

+ [1 (1 − 𝜆2) + 𝜉2𝜆2] 𝑒−𝑟𝑑

(1 − 𝜋) (1 − 𝜆1)

+𝜉1𝜆1} 𝑒−𝑟0

.

(10)

When 𝜆1, 𝜆2can be got by the above formula, with the same

method, we can get the risk interest rate {𝜆𝑖, 𝑖 ≥ 1} of each

period.

4 Abstract and Applied Analysis

𝑟0 + 𝜇0 + 𝜎0

𝑟0 + 𝜇0 − 𝜎0

𝑟0 +1∑𝑗=0𝜇𝑗 + 2𝜎1

𝑟0 +1∑𝑗=0𝜇𝑗

𝑟0 +1∑𝑗=0𝜇𝑗 − 2𝜎1

𝑟0 +2∑𝑗=0𝜇𝑗 + 3𝜎2

𝑟0 +2∑𝑗=0𝜇𝑗 + 𝜎2

𝑟0 +2∑𝑗=0𝜇𝑗 − 𝜎2

𝑟0 +2∑𝑗=0𝜇𝑗 − 3𝜎2

𝑟0

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

𝜋

1 − 𝜋

Figure 2: 4-period time-varying interest rate tree.

3. Stock and Interest Rate Binary Tree Modelwith Credit Risk

For convertible bonds with credit risk, suppose that theunderlying stock price and risk-free interest rate process arerandom, and the underlying stock price process is describedby CRR model, where the stock magnitude of up and downis 𝑢 = 𝑒

𝜎𝑆√Δ𝑡, 𝑑 = 𝑒

−𝜎𝑆√Δ𝑡 respectively. Suppose that the stock

price is 0 when default, and then the possible price of stock is0, 𝑆𝑢, 𝑆𝑑.

In risk-neutral world, the expected yield rate is risk-freeinterest rate 𝑟, and the stock continuous dividends yield is 𝑞,then the expected yield rate is 𝑟−𝑞; so tomeet the no-arbitragecondition, there is

𝑆𝑒(𝑟−𝑞)Δ𝑡

= 𝑝 (1 − 𝜆) 𝑆𝑢 + (1 − 𝑝) (1 − 𝜆) 𝑆𝑑 + 0 ⋅ 𝜆. (11)

Then get 𝑝 = (𝑒(𝑟−𝑞)Δ𝑡

/((1 − 𝜆) − 𝑑))/(𝑢 − 𝑑). 𝑝 is the upprobability of stock with credit risk. As the risk-free interestrate of all periods is random, suppose that the risk-freeinterest rate of number 𝑖 period is 𝑟

𝑖, the volatility of stock

is constant 𝜎𝑆, and dividends rate is 𝑞

𝑖, so the parameters of

stock price in all periods can be generally presented as

𝑢𝑖= 𝑒𝜎𝑆√Δ𝑡

, 𝑑𝑖= 𝑒−𝜎𝑆√Δ𝑡

,

𝑝𝑖=

𝑒(𝑟𝑖−𝑞𝑖)Δ𝑡/ (1 − 𝜆

𝑖) − 𝑑

𝑢 − 𝑑

.

(12)

Suppose risk-free interest rates are stochastic and describedby binary tree model, then the stock tree and interest ratetree involving credit risk are combined as shown in Figure 3.In this paper, we suppose that the correlation coefficient ofinterest rate and stock price is 0.

After obtaining the process of stock prices and risk-freeinterest rates, the value of convertible bonds can be got bybackward induction.Wedivide the value of convertible bondsinto two parts; one is the value of equity got by converting tostock or exercise embedded options; the other is bonds value

𝜉1

𝜆1

𝑟, 𝑆

𝑟𝑢, 𝑆𝑢

𝑟𝑑, 𝑆𝑢

𝑟𝑢, 𝑆𝑑

𝑟𝑑, 𝑆𝑑

𝜋𝑝1(1 − 𝜆1)

(1 − 𝜋)𝑝1(1 − 𝜆1)

(1 − 𝜋)(1 − 𝑝1)(1 − 𝜆1)

𝜋(1 − 𝑝1)(1 − 𝜆1)

Figure 3: 4-period two-factor binary tree with credit risk added.

that is the present value of bonds when repaying capital andinterest and the present value of the residual value.

Suppose that the default probability of convertible bondsin the interval [𝑡

𝑘−1, 𝑡𝑘] is 𝜆𝑘, recovery value is 𝜉

𝑘, and then

the holding value at 𝑡𝑘time is 𝐸𝑉

𝑘, given by

𝐸𝑉𝑘= (the expected equity value at 𝑡

𝑘+1-time

+ the expected bonds value at

𝑡𝑘+1

-time) ⋅ 𝑒−𝑟𝑘 ⋅Δ𝑡.

(13)

So the value of convertible value at time-𝑡𝑘is

𝑉𝑘= max [min (the holding value at

𝑡𝑘-time, call value) ,

conversion value, put value]

= max [min (𝐸𝑉𝑘, 𝑉

call𝑘

) , 𝑉con𝑘

, 𝑉put𝑘

] .

(14)

Abstract and Applied Analysis 5

4. Numerical Examples

We take a four-period binary tree model as an exampleto expound the convertible bonds pricing process with callprovision and put provision in the abovemodels and comparethe results under the constant volatility interest rate modeland the time-varying volatility interest rate model.

4.1. Process of Interest Rate and Stock. The initial parametersof the convertible bond are all the same to the constantvolatility interest rate model and the time-varying volatilityinterest rate model. Suppose that time interval Δ𝑡 = 1, theup probability of interest rates is 𝜋 = 1/2, and the 1–4-year period yields of risk-free zero-coupon bonds are 6.145%,6.366%, 6.837%, and 6.953%, respectively; the volatility ofshort-term interest rates is 2.5%. So the other binary treeparameters of constant volatility interest rate can be got asshown in Table 1. In the same way, the other binary treeparameters of time-varying volatility interest rate can be gotas shown in Table 2. Then we get the two interest rate binarytrees.

The process of underlying stock prices uses CRR modelto describe.The selected parameters 8 are as follows: 𝑆

0= 25,

𝜎𝑆= 0.185, Δ𝑡 = 1, 𝑇 = 4, and 𝑞 = 0.04. So all the parameters

under constant volatility interest rate are

𝑢 = 1.2032, 𝑑 = 0.8311,

𝑝𝑟0

= 0.5887, 𝑝𝑟(1)𝑢

= 0.6841,

𝑝𝑟(1)𝑑

= 0.5923, 𝑝𝑟(2)𝑢𝑢

= 0.7517,

𝑝𝑟(2)𝑢𝑑

= 0.6577, 𝑝𝑟(2)𝑑𝑑

= 0.5667,

𝑝𝑟(3)𝑢𝑢𝑢

= 0.9162, 𝑝𝑟(3)𝑢𝑢𝑑

= 0.8170,

𝑝𝑟(3)𝑢𝑑𝑑

= 0.7209, 𝑝𝑟(3)𝑑𝑑𝑑

= 0.6279.

(15)

In the sameway, the parameters under time-varying volatilityinterest rate model are

𝑢 = 1.2032, 𝑑 = 0.8311,

𝑝𝑟0

= 0.5887, 𝑝𝑟(1)𝑢

= 0.6826,

𝑝𝑟(1)𝑑

= 0.5908, 𝑝𝑟(2)𝑢𝑢

= 0.7475,

𝑝𝑟(2)𝑢𝑑

= 0.6594, 𝑝𝑟(2)𝑑𝑑

= 0.5739,

𝑝𝑟(3)𝑢𝑢𝑢

= 0.8764, 𝑝𝑟(3)𝑢𝑢𝑑

= 0.8027,

𝑝𝑟(3)𝑢𝑑𝑑

= 0.7307, 𝑝𝑟(3)𝑑𝑑𝑑

= 0.6604.

(16)

Then we get the four-period stock prices binary tree.

4.2. Default Rates. We take the corporate bonds as refer-ence risk bonds; suppose that the 1–4-year period yields ofcorporate zero-coupon bonds are 7.645%, 8.155%, 8.557%,and 9.128%, respectively, and the recovery rate of convertiblebonds is constant 𝜉 = 45%, and one-year risk-free interestrate 𝑟0= 6.145%.

Interest rate binary tree indicates that the branch point ofnumber 𝑛 period is 𝑛. If the interest rate of number 𝑖 branchpoint in number 𝑛 period is 𝑟(𝑛 − 1)

𝜔, 𝜔 is the interest rates

state from start to current, and then the derived interest ratebranch point of number 𝑛 + 1 period is 𝑟(𝑛)

𝜔𝑢, 𝑟(𝑛)𝜔𝑑. As 𝜋 =

1/2 = 1 − 𝜋, to meet the no-arbitrage principle, parameters{𝜆1, 𝜆2, 𝜆3, 𝜆4}meet the following four equations:

𝑒−𝑟∗

1+𝑟0

= (1 − 𝜆1) + 𝜉 ⋅ 𝜆

1,

𝑒−2𝑟∗

2+𝑟0

= 𝜋 (1 − 𝜆1) (1 − 𝜆

2+ 𝜉𝜆2)

× (𝑒−𝑟(1)𝑢

+ 𝑒−𝑟(1)𝑑

) + 𝜉𝜆1,

𝑒−3𝑟∗

3+𝑟0

= 𝜋2(1 − 𝜆

1) (1 − 𝜆

2) (1 − 𝜆

3+ 𝜉𝜆3)

× [𝑒−𝑟(1)𝑢

(𝑒−𝑟(2)𝑢𝑢

+ 𝑒−𝑟(2)𝑢𝑑

)

+𝑒−𝑟(1)𝑑

⋅ (𝑒−𝑟(2)𝑢𝑑

+ 𝑒−𝑟(2)𝑑𝑑

)]

+ 𝜋 (1 − 𝜆1) 𝜉𝜆2(𝑒−𝑟(1)𝑢

+ 𝑒−𝑟(1)𝑑

) + 𝜉𝜆1,

𝑒−4𝑟∗

4+𝑟0

= 𝜋3(1 − 𝜆

1) (1 − 𝜆

2) (1 − 𝜆

3) (1 − 𝜆

4+ 𝜉𝜆4)

× {𝑒−𝑟(1)𝑢

[𝑒−𝑟(2)𝑢𝑢

(𝑒−𝑟(3)𝑢𝑢𝑢

+ 𝑒−𝑟(3)𝑢𝑢𝑑

)

+𝑒−𝑟(2)𝑢𝑑

(𝑒−𝑟(3)𝑢𝑢𝑑

+ 𝑒−𝑟(3)𝑢𝑑𝑑

)]

+ 𝑒−𝑟(1)𝑑

[𝑒−𝑟(2)𝑢𝑑

(𝑒−𝑟(3)𝑢𝑢𝑑

+ 𝑒−𝑟(3)𝑢𝑑𝑑

)

+𝑒−𝑟(2)𝑑𝑑

(𝑒−𝑟(3)𝑢𝑑𝑑

+ 𝑒−𝑟(3)𝑑𝑑𝑑

)]}

+ 𝜋2(1 − 𝜆

1) (1 − 𝜆

2) 𝜉𝜆3

× [𝑒−𝑟(1)𝑢

(𝑒−𝑟(2)𝑢𝑢

+ 𝑒−𝑟(2)𝑢𝑑

)

+𝑒−𝑟(1)𝑑

(𝑒−𝑟(2)𝑢𝑑

+ 𝑒−𝑟(2)𝑑𝑑

)]

+ 𝜋 (1 − 𝜆1) 𝜉𝜆2⋅ (𝑒−𝑟(1)𝑢

+ 𝑒−𝑟(1)𝑑

) + 𝜉𝜆1.

(17)

By the above equations and the constant volatility interestrate binary tree, the default rates of bonds in every period areshown in Table 3.

Similarly, the default rates of corporate bonds in everyperiod under time-varying volatility interest rate binary treeare shown in Table 4.

4.3. Price Process of Convertible Bonds. The convertible bondcontains call and put provisions, the duration is 𝑇 = 4, theface value got in maturity date is 100, conversion rate is 3,callable price is 𝑉call = 106, and puttable price is 𝑉put = 80.We suppose that the investors can exercise the puttable rightafter one year.

Now we take the convertible bond under time-varyingvolatility interest rate binary tree to explain its pricingprocess. Take four points A, B, C, and D in pricing tree ofFigure 4 into consideration; among them, C, D are at theend of period, 4, B are at the end of period 3, and A is at

6 Abstract and Applied Analysis

Table 1: Constant volatility interest rate parameters.

Deadlineyear(s)𝑡

Price ofbonds (yuan)

𝑃(𝑡)

Volatility ofshort-term

interest rate (%)𝜎(𝑡)

Annualprofit rate ofbonds (%)

𝑦(𝑡)

1-yearlong-terminterest rate

(%)𝑓(𝑡)

Variance𝜎2(∑𝑡

𝑗=1𝑟(𝑗))

Sum ofDelta (%)∑𝑡−1

𝑗=1𝛿(𝑗)

Delta(%)𝛿(𝑡)

Drift item(%)𝜇(𝑡)

Sum of driftitems (%)∑𝑡−1

𝑗=0𝜇(𝑗)

Expectation(%)

𝐸𝑄

0[𝑟(𝑡)]

0 1.6 6.145 0.0128 0.0128 0.4548 0.4548 6.1451 0.9404 6.145 6.587 0.000256 0.0512 0.0384 1.2304 1.6852 6.59982 0.8805 6.366 7.779 0.00128 0.1152 0.064 −0.414 1.2712 7.83023 0.8146 6.837 7.301 0.003584 0.2048 0.0896 7.41624 0.7572 6.953 0.00768

Table 2: Time-varying volatility interest rate parameters.

Deadlineyear(s)𝑡

Price ofbonds (yuan)

𝑃(𝑡)

Volatility ofshort-term

interest rate (%)𝜎(𝑡)

Annualprofit rate ofbonds (%)

𝑦(𝑡)

1-yearlong-terminterest rate(%) 𝑓(𝑡)

Variance𝜎2(∑𝑡

𝑗=1𝑟(𝑗))

Sum ofDelta (%)∑𝑡−1

𝑗=1𝛿(𝑗)

Delta(%)𝛿(𝑡)

Drift item(%)𝜇(𝑡)

Sum of driftitems (%)∑𝑡−1

𝑗=0𝜇(𝑗)

Expectation(%)

𝐸𝑄

0[𝑟(𝑡)]

0 1.6 6.145 0.0128 0.0128 0.4548 0.4548 6.1451 0.9404 1.5 6.145 6.587 0.000256 0.0465 0.0337 1.2257 1.6805 6.59982 0.8805 1.2 6.366 7.779 0.001186 0.0768 0.0303 −0.4477 1.2328 7.82553 0.8146 1.3 6.837 7.301 0.002722 0.1404 0.0636 7.37784 0.7572 6.953 0.00553

Table 3: Default rate of corporate bonds under constant volatilityinterest rate model.

Time period 0-1 (𝜆1) 1-2 (𝜆

2) 2-3 (𝜆

3) 3-4 (𝜆

4)

Default rate 0.0271 0.0394 0.0342 0.0737

Table 4: Default rate of corporate bonds under time-varyingvolatility interest rate model.

Time period 0-1 (𝜆1) 1-2 (𝜆

2) 2-3 (𝜆

3) 3-4 (𝜆

4)

Default rate 0.0271 0.0389 0.0348 0.0734

the end of period 2. At point C, as the convertible valueis 108.57 that is larger than the face value 100, the value ofconvertible bond is 108.57, while the bonds value is 0, sowrite it as [108.57, 0]; the first numbermeans the equity valueand the second number means the bonds value. In the sameway, we can get the convertible value of D point that canbe written as [0, 100]. And the up probability of B point is𝑝𝑟(3)𝑢𝑢𝑑

(1−𝜆4) = 0.8027× (1−0.0734) = 0.7438. At the same

way, the down probability is 0.1828.Then the equity value of Bpoint is 108.57 × 0.7438 × 𝑒

−0.08578= 74.12. The bonds value

of B point is (45 × 0.0734 + 100 × 0.1828)𝑒−0.08578

= 19.81.Then the holding value 𝐸𝑉

𝐵is 93.93. And the convertible

value at B point is 90.24, so the value of convertible bondsin B point is

𝑉𝐵= max [min (𝐸𝑉

𝐵, 𝑉call) , 𝑉

con𝐵

, 𝑉put]

= max [min (93.93, 108) , 90.24, 80] = 93.93,

(18)

written as [74.12, 19.81].

In the same way, we can calculate the value of other threebranch points E, F, and G at the end of period 3 that are[125.51, 2.96], [125.51, 3.03], and [79.00, 13.22]. So the equityvalue at A point is

(125.51 × 0.3607 + 125.51 × 0.3607 + 79.00

× 0.1219 + 74.12 × 0.1219) × 𝑒−0.10826

= 98.00.

(19)

The bonds value at A point is

(45 × 0.0348 + 2.96 × 0.3607 + 3.03 × 0.3607

+13.22 × 0.1219 + 19.81 × 0.1219) 𝑒−0.10826

= 6.96.

(20)

Then the holding value 𝐸𝑉𝐴is 104.96. And the convertible

value at A point is 108.57, so the value of convertible bonds inA point is

𝑉𝐴= max [min (𝐸𝑉

𝐴, 𝑉call) , 𝑉

con𝐴

, 𝑉put]

= max [min (104.96, 108) , 108.57, 80] = 108.57,

(21)

written as [108.57, 0]. The other branch point in the pricingbinary tree of convertible bonds can be got in the same way.

At last, under time-varying volatility interest rate binarytree model, we can get the price of convertible bond whichcontains credit risk that is 79.32 at the time of 𝑡 = 0. Similarly,under constant volatility interest rate binary tree model, wecan get the price of convertible bond which contains creditrisk that is 78.52 at the time of 𝑡=0; this is less than the former.

Abstract and Applied Analysis 7

𝜉𝐹 = 45

𝜉𝐹 = 45

𝜉𝐹 = 45

𝜉𝐹 = 45

𝜉𝐹 = 45

𝜆4

𝜆4

𝜆4

𝜆4

𝜆3

𝑝𝑟(3)𝑢𝑢𝑢 (1 − 𝜆4)

𝑝𝑟(3)𝑢𝑢𝑑 (1 − 𝜆4)

𝑝𝑟(3)𝑢𝑢𝑢 (1 − 𝜆4)

𝑝𝑟(3)𝑢𝑢𝑑 (1 − 𝜆4)

𝑆(4)𝑢𝑢𝑢𝑢 = 52.39

𝑆(4)𝑢𝑢𝑢𝑢 = 52.39

𝑆(4)𝑢𝑢𝑢𝑑 = 36.19

𝑆(4)𝑢𝑢𝑢𝑑 = 36.19

𝑆(4)𝑢𝑢𝑑𝑑 = 25

𝑆(4)𝑢𝑢𝑑𝑑 = 25

𝑆(4)𝑢𝑢𝑢𝑑 = 36.19

𝑆(4)𝑢𝑢𝑢𝑑 = 36.19

A

BC

D

E

F

G

𝑟(3)𝑢𝑢𝑢 = 0.10978𝑆(3)𝑢𝑢𝑢 = 43.54

𝑟(3)𝑢𝑢𝑑 = 0.08578𝑆(3)𝑢𝑢𝑢 = 43.54

𝑟(2)𝑢𝑢 = 0.10826𝑆(2)𝑢𝑢 = 36.19

𝑟(3)𝑢𝑢𝑢 = 0.10978𝑆(3)𝑢𝑢𝑑 = 30.08

𝑟(3)𝑢𝑢𝑑 = 0.08578𝑆(3)𝑢𝑢𝑑 = 30.08

(1 − 𝑝𝑟(3)𝑢𝑢𝑢 )(1 − 𝜆4)

(1 − 𝑝𝑟(3)𝑢𝑢𝑑 )(1 − 𝜆4)

(1 − 𝑝𝑟(3)𝑢𝑢𝑢 )(1 − 𝜆4)

(1 − 𝑝𝑟(3)𝑢𝑢𝑑 )(1 − 𝜆4)

Figure 4: Constructing pricing tree under time-varying volatility interest rate model.

5. Conclusions

Binary tree method is a classical pricing method, by con-structing the binary tree of state variable to describe thepossible paths of state variable in the duration of contingentclaims and then tomake pricing research. Binary treemethodcan effectively solve the path-dependent options pricing,intuitive and easy to operate. As the embedded optionsin the convertible bonds are all American options, binarytree method becomes one of the main pricing methods ofconvertible bonds. Interest rate is the main factor whichimpacts the price of convertible bonds; the description ofits binary tree model is the main problem of convertiblebonds pricing.This paper adopts constant volatility and time-varying volatility binary tree model to describe interest ratesand further consider the impact of stock dividends and creditrisk to the price of convertible bonds, adopt default rate andrecovery rate to describe the credit risk, and get the two-factor binary treemodel with credit risk added. Based on this,we make a numerical example and get the convertible bondspricing result under the stock prices obeying CRRmodel andthe constant and time-varying volatility interest rate binarytree model. The model can be popularized to the pricing ofconvertible bonds with more complex provisions and otherfinancial derivatives such as bond options, catastrophe bonds,and mortgage-backed security.

AcknowledgmentsThis work was supported in part by the Natural ScienceFoundation of China (no. 71071166, no. 71201013, and no.70921001) and the Ministry of Education of Humanities andSocial Science Project of China (no. 12YJC630118).

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